Fourier-series and periods

1. The problem statement, all variables and given/known data
I have the Fourier-series for the function:

f(x) = x for -Pi < x < Pi.

I know wish to find the Fourier-series for the function:

g(x) = x for -p < x < p,

where p is a random period.

3. The attempt at a solution
Ok, the obvious thing here is to use the result from f(x). If thought that if I define f(p*x/Pi) = g(x) = f(y), then I just insert x=y*Pi/p. But in the result of this exercise, they have an additional p/Pi in front of the sum. Where does this come from?